States and Structure of von Neumann Algebras - Springer Link

C 2004) International Journal of Theoretical Physics, Vol. 43, Nos. 7/8, August 2004 (

States and Structure of von Neumann Algebras Jan Hamhalter1

We summarize and deepen recent results on the interplay between properties of states and the structure of von Neumann algebras. We treat Jauch–Piron states and the concept of independence in noncommutative probability theory. KEY WORDS: states on von Neumann algebras; Jauch–Piron states; independence of algebras.

1. JAUCH–PIRON STATES, σ-ADDITIVITY, AND FACIAL STRUCTURE In this part we analyze connection between the Jauch–Piron property, σ -additivity of states and facial structure of duals of von Neumann algebras. (For basic facts on von Neumann algebras we refer to Kadison and Ringrose, 1983.) Let M be a von Neumann algebra with the projection lattice P(M). A state  (i.e. a positive normalized functional) on M is called Jauch–Piron if (e ∨ f ) = 0 whenever e and f are projections in M with (e) = ( f ) = 0. (The symbol e ∨ f stands for the supremum of projections e, f .) In the physical interpretation, projection e represents a random event of the system given by the algebra M. The linear functional  describes probability distributions of all observables. The value (e) is probability of event e in state  of the system. The Jauch–Piron property now postulates that the events with zero probability obey the same law as in classical probability theory: if events e and f have both probability zero then the probability that event e or f occurs is again zero. Not all states on von Neumann algebras have this property (Amann, 1989; Bunce and Hamhalter, 1994). That is why the concept of Jauch–Piron state has received a great deal of attention. It was discovered and first studied by Jauch and Piron (Jauch, 1968; Jauch and Piron, 1965, 1969) in connection with propositional calculus of quantum mechanics and the theory of hidden variables. The first results on Jauch–Piron states on operator algebras were obtained by G. R¨utimann (1977). A nice characterization of pure 1

Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University, 166 27 Prague 6, Czech Republic; e-mail: [email protected] 1561 C 2004 Springer Science+Business Media, Inc. 0020-7748/04/0800-1561/0 

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Jauch–Piron states on algebras on separable spaces has been given by A. Amann (1989). We have characterized the Jauch–Piron property in terms of functional analytic properties of states on von Neumann algebras and JW algebras in a series of papers (Bunce and Hamhalter, 1994, 1996, 2000; Hamhalter, 1993). This study has revealed an interesting fact that continuity properties of Jauch–Piron states are intimately connected with the lattice-theoretic properties of the projection lattice. In this way, the concept of Jauch–Piron state, introduced solely on physical grounds, has also produced a harvest of results for pure operator algebra theory. In this section, we would like to comment on the relationship between σ -additivity of states and the Jauch–Piron A state  on a von Neumann algebra M is  property. ∞ said to be σ -additive if ( ∞ e ) = n n=1 n=1 (en ), whenever (en ) is a sequence of orthogonal projections in M. If M is σ -finite (i.e. if any system of mutually orthogonal projections in M is at most countable) then  is σ -additive if and only if  is normal. (A state is called normal if it is continuous with respect to w ∗ -topology.) By using the spectral theorem, it is not difficult to show that any σ -additive state is Jauch–Piron (Bunce and Hamhalter, 1994). On the other hand, the reverse implication is far from being true. For example, let M be a von Neumann algebra acting on a separable infinite-dimensional Hilbert H . Given an orthonormal basis  space 1 (ξn ) of H we can define a state 1 = ∞ n=1 2n ωn , where ωn (x) = (xξn , ξn ) for all x ∈ M. It can be easily verified that the state 1 is σ -additive and faithful. (A faithful state attains nonzero value on each nonzero positive element.) As H has infinite dimension, we can find a state 2 vanishing on all one-dimensional projections. Consider now the state  = 12 (1 + 2 ). The state  is again faithful and thereby Jauch–Piron. Nevertheless,  is not σ -additive  because for one-dimensional projections en projecting onto sp{ξn } we get, ∞ n=1 en = 1, and so       ∞ ∞ ∞    1 1 1= en =  1 en +  2 en 2 2 n=1 n=1 n=1 =

∞ ∞ ∞  1 1 1 1 (en ) + = 1 (en ) = (en ). 2 n=1 2 2 n=1 n=1

Another natural example is given by a tracial state. A tracial state  is a state satisfying (x ∗ x) = (x x ∗ ) for all x ∈ M. Any trace is Jauch–Piron, but it need not be σ -additive in general (Bunce and Hamhalter, 1994). So examples of non-σ additive Jauch–Piron states are generic. However, if we combine the Jauch–Piron condition with some geometric properties of states, we can enforce σ -additivity. A prototype result of this type, proved by A. Amann (1989) in separable case and in (Bunce and Hamhalter, 1994) in general case, is the following theorem. Theorem 1.1. Let M be a von Neumann algebra without nonzero abelian direct summand. A pure state on M is Jauch–Piron if and only if it is σ -additive.

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In order to extend this result, we introduce a few concepts. Denote by S(M) the convex set of all states on a von Neumann algebra M. We shall call S(M) the state space of M. Endowed with w ∗ -topology S(M) is compact. A subset F ⊂ S(M) is called face if F is convex and 1 , 2 ∈ F whenever 12 (1 + 2 ) ∈ F. Singleton set {} is a face if and only if  is a pure state. Moreover, we say that a face F is a split face if there is a face F # (called complementary face) such that each  ∈ S(M) is a unique convex combination of states 1 ∈ F and 2 ∈ F # . There are deep results characterizing faces of the state space in terms of projections in the double dual of the algebra in question (see e.g. Alfsen and Schultz, 1976). All norm closed faces of S(M) are of the form { ∈ S(M)|( p) = 1}, where p is a projection in the double dual M ∗∗ . All split faces are of the form { ∈ S(M)|(z) = 1}, where z is a central projection in M ∗∗ . The following examples of split faces will be important in the sequel: Sσ (M), the set of all σ -additive states, and Sn (M), the set of all normal states. We shall study the position of the set SJ (M) of all Jauch–Piron states on M and the split face Sσ (M). We have already seen that Sσ ⊂ SJ (M). It can be easily verified that SJ (M) is a convex set and that any extreme point of SJ (M) is simultaneously an extreme point of the whole state space, i.e. a pure state (Bunce and Hamhalter, 2000). However, unlike Sσ (M) the set SJ (M) is only exceptionally norm closed or a split face as the following theorem shows. Theorem 1.2. (Bunce and Hamhalter (2000)) For a von Neumann algebra M the following statements are equivalent: (i) (ii) (iii) (iv)

S J (M) is a face. S J (M) is norm closed. S J (M) = S(M) (all states are Jauch–Piron). M = M1 ⊕ M2 , where M1 is abelian von Neumann algebra and M2 has finite dimension.

It demonstrates how different is the σ -additivity and the Jauch–Piron property. If SJ (M) is a proper subset of S(M) then SJ (M) is never a face and it is never norm closed. Surprisingly, making norm closure of SJ (M) we always get even split face. Theorem 1.3. (Bunce and Hamhalter (2000)) Let M be a von Neumann algebra. Then S J (M) is a split face. Its complementary face #

S J (M) ⊂ { ∈ S(M)|(e) = 0

for all σ -finite projections e}.

The second part of the previous theorem says that the complementary face of SJ (M) consists of extremely discontinuous states. This is due to the fact that any projection is a sum of orthogonal σ -finite subprojections. In other words, the closure of the Jauch–Piron state space contains all reasonably continuous states.

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For example, the set of all Jauch–Piron states is already dense in the whole state space provided that the algebra is σ -finite. The analysis of the facial structure of the Jauch–Piron state space has led to the following criterion of σ -additivity in terms of the Jauch–Piron property. Given a state  on M and a ∈ M with (a ∗ a) = 0 we define the transformed state a by setting a (x) =

(a ∗ xa) , (a ∗ a)

x ∈ M.

In case of M being abelian, the set of all transformed states of  corresponds to the set of all measures absolutely continuous with respect to . Theorem 1.4. (Bunce and Hamhalter (2000)) Let  be a state on a von Neumann algebra M with no nonzero abelian direct summand. The following conditions are equivalent: (i)  is σ -additive. (ii) The norm closure of the set of transformed states of  consists of Jauch–Piron states. (iii) The split face generated by  consists of Jauch–Piron states. Thus, an individual Jauch–Piron state need not be σ -additive. However, if we require that all states obtained by manipulating with the physical system initially prepared in a state  are Jauch–Piron, then  has to be σ -additive. This may be seen as justification of σ -additivity on physical grounds. By choosing  to be a pure state Theorem 1.4 reduces to Theorem 1.1. Indeed, the split face generated by a pure state  is the set of all unitary transforms of . As the unitary transform preserves the Jauch–Piron property we see by equivalence of (i) and (iii) in the previous Theorem that Theorem 1.1 holds. Besides, Theorem 1.4 has the following consequence, saying that the set Sσ (M) is uniquely determined by the Jauch–Piron state space SJ (M). Corollary 1.5. (Bunce and Hamhalter (2000)) If M is a von Neumann algebra with no nonzero abelian direct summand then Sσ (M), the split face of all σ -additive states on M, is the largest split face contained in S J (M). Another result showing that σ -additivity is determined by the structure of Jauch–Piron states is the following result. (By the symbol π ∗ we mean the adjoint map of a linear map π .) Theorem 1.6. Let M and N be von Neumann algebras, M having no nonzero abelian direct summand. Let π : M → N be a ∗-homomorphism. Then π is

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σ -additive if and only if π ∗ (SJ (N )) ⊂ SJ (M).

(1)

Proof: It has been proved in (Bunce and Hamhalter, 1995) that π is σ -additive if and only if π is a lattice homomorphism i.e. π (e ∨ f ) = π (e) ∨ π( f )

f or all pr ojections e, f ∈ M.

(2)

We show that (2) is equivalent to (1). Suppose that π has the property (1). It means that ϕ ◦ π is a Jauch–Piron state on M whenever ϕ is a Jauch–Piron state on N . Assume, for a contradiction, that π (e ∨ f ) = π (e) ∨ π ( f ) for some projections e, f ∈ P(M). As π (e) ∨ π ( f ) ≤ π (e ∨ f ) it means that the projection p = π(e ∨ f ) − (π(e) ∨ π ( f )) is nonzero. Therefore, we can find a normal state ψ with ψ( p) = 1. It implies that ψ(π (e) ∨ π ( f )) = 0 and so ψ(π(e)) = ψ(π( f )) = 0. The state ψ is normal and therefore Jauch–Piron. Nevertheless, for the state ψ ◦ π we have (ψ ◦ π )(e ∨ f ) = 1

while ψ ◦ π (e) = ψ ◦ π ( f ) = 0,

contradicting (1). The implication (2)=⇒(1) is obvious. The proof is completed.
We now summarize our discussion in a form of a few equivalent conditions characterizing the σ -additivity in terms of physically plausible conditions. Theorem 1.7. Let  be a state on a von Neumann algebra M having no nonzero abelian direct summand. We denote by π the G.N.S. representation of  and by M the von Neumann algebra generated by π (M). The following conditions are equivalent. (i)  is σ -additive. (ii) The norm closure of the set of all transformations of  consists of Jauch–Piron states. (iii) The G.N.S. representation π is σ -additive. (iv) ϕ ◦ π is a Jauch–Piron state whenever ϕ is a Jauch–Piron state on M . (v) ϕ ◦ π is a Jauch–Piron state whenever ϕ is a normal state on M . (vi) ϕ ◦ π is a Jauch–Piron state whenever ϕ is a σ -additive state on M . (vii) ϕ ◦ π is a Jauch–Piron state whenever ϕ is a vector state on M . Proof: Equivalence of conditions (i) and (ii) has been established earlier. Let us verify that (i) is equivalent to (iii). It is obvious that  is σ -additive provided that π is σ -additive. Conversely, suppose that  is σ -additive. Let (en )

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be an increasing sequence of projections in M such that en  e. Let (H , π , h  ) be G.N.S. data of , i.e. h  ∈ H is a unit vector such that (a) = (π (a)h  , h  )

for all a ∈ M.

For each unitary u ∈ M one gets (π (en )π (u)h  , π (u)h  ) = (u ∗ en u)  (u ∗ eu) = (π (e)π (u)h  , π (u)h  ) as n → ∞. In other words, as π (e − en ) is a projection, we obtain ||π (e − en )π (u)h  2 = (π (e − en )π (u)h  , π (u)h  ) = (u ∗ eu) − (u ∗ en u) → 0 as n → ∞.

(3)

As M is a linear span of its unitary group, the linear span of the set {π (u)h  |u is unitary} is dense in H . Hence, by (3) π (e − en )h → 0 for each h ∈ H . It means that (π (en )) converges to π (e) in the strong operator topology and so π (en )  π (e). Thus, π is σ -additive. The equivalence of conditions (iv)–(vii) can be derived in the same way as in the proof of Theorem 1.6. The proof is completed.
The result on σ -additivity of Jauch–Piron pure states which was a starting point of our discussion can also be extended to much general states called factor states. A state  on a von Neumann algebra M is called a factor state if the weak operator closure of π (M) is a factor, where π is a G.N.S. representation of . The set of factor states, sometimes called the factor spectrum, plays an important role in the structure theory of von Neumann algebras and comprises e.g. pure states and special convex combinations of pure states. Factor states are characterized as being concentrated on atomic central projections in the double dual: a state  on M is a factor state if and only if (z) = 1 for some minimal central projection z in M ∗∗ . It has turned out that more or less all factor states on properly infinite von Neumann algebras are σ -additive. Theorem 1.8. (Bunce and Hamhalter (2000)) Let M be a properly infinite von Neumann algebra. Suppose that M is σ -finite or the continuum hypothesis is true. Then all Jauch–Piron states are σ -additive. This result does not hold for finite algebras. For example, let M = Z ⊗Mn be a tensor product of an infinite-dimensional abelian von Neumann algebra Z and the algebra Mn of all n × n matrices. Let  = 1 ⊗ τ , where τ is the normalized trace and 1 is a singular state. (A state is called singular if it belongs to the complementary split face Sn# (M).) Then  is an example of non-σ -additive Jauch– Piron type I factor state. (The factor state is called type I if the algebra generated

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by its G.N.S. representation is a type I factor.) In this regard, the criterion of σ -additivity in Theorem 1.4 can be simplified as follows. Proposition 1.9. Let  be a type I factor state on a von Neumann algebra M. Then  is σ -additive if and only if the split face F generated by  contains at least one Jauch–Piron state which is an extreme point of the set SJ (M). Proof: Suppose that ϕ ∈ F is an extreme point of SJ (M). By (Bunce and Hamhalter, 2000) ϕ is a pure state, which implies that ϕ is σ -additive (Theorem 1.1). So the split face F has nonzero intersection with the split face Sσ (M). As the split faces are represented by central projections in M ∗∗ we can write F = {ψ ∈ S(M)|ψ(z  ) = 1}, Sσ (M) = {ψ ∈ S(M)|ψ(z σ ) = 1}, where z  and z σ is a minimal central projection and and a central projection in M ∗∗ , respectively. So either z  ≤ z σ or z σ z  = 0. As the second possibility is excluded by the assumption we see that  ∈ F ⊂ Sσ (M). On the other hand, if  is σ -additive then F consists of σ -additive states and it contains a pure state because π (M) ∼ = z  M ∗∗ is a type I factor.
The proof is complete. An analysis of the Jauch–Piron property for factor states on homogeneous type I algebras has led to the following result: Theorem 1.10. (Bunce and Hamhalter (2000)) Let M = C(X ) ⊗ Mn , where 2 ≤ n < ∞, C(X ) is algebra of continuous functions on compact hyperstonean space, and let  be a factor state on M: (i) If  is a Jauch–Piron state and is not faithful on the center of M, then  is σ -additive. (ii) If  is faithful on the center of M, then  is a Jauch–Piron state. Theorems 1.8 and 1.10 demonstrate the fact that for highly noncommutative von Neumann algebras the algebraic and functional analytic properties of states are better connected then in case of algebras retaining some properties of abelian algebras.

2. INDEPENDENCE OF VON NEUMANN ALGEBRAS In this section, we would like to consider the question of independence of von Neumann algebras. In classical probability theory, two probability spaces

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(X, A1 , µ1 ) and (Y, A2 , µ2 ) living in a probability space (Z , A, µ) are called independent if (Z , A, µ) can be organized as the product space (X × Y, A1 × A2 , µ1 × µ2 ). Especially, for independent random variables f and g corresponding to systems X and Y , respectively, we have well-known principle for expectation value E : E( f g) = E( f )E(g). The independence is a well-established concept in classical probability theory. Unlike this, there are many mutually nonequivalent generalization of this notion in noncommutative probability theory arising mostly in quantum filed theory (Baumg¨artel and Wollenberg, 1992; Baumg¨artel, 1995; Goldstein et al., 1999; Florig and Summers, 1997; Haag, 1996; Haag and Kastler, 1964; Hamhalter, 1997, 1998, 2001, to appear; Horudzij, 1990; Kraus, 1995; Redei, 1995a,b, 1998; Roos, 1970; Summers, 1988, 1990). In our discussion, we shall focus mainly on extension type independence conditions and their interrelations. Let A and B be von Neumann subalgebras in a von Neumann algebra M. (We will always assume that subalgebra contains the unit of larger algebra.) The pair (A, B) is said to be C ∗ -independent if for any state ϕ1 on A and for any state ϕ2 on B there is a state ϕ on M extending both ϕ1 and ϕ2 . This concept was proposed by Haag and Kastler (1964) in the context of axiomatic quantum field theory. An analogous notion in the category of W ∗ -algebras is the W ∗ -independence. The pair (A, B) is said to be W ∗ -independent if for normal states ϕ1 and ϕ2 on A and B, respectively, there is a normal state ϕ on M extending both ϕ1 and ϕ2 . In classical measure theory, the W ∗ -independence amounts to a question of the existence of a common completely additive extension of two completely additive measures prescribed on subsystems. The problem of the relationship between C ∗ - and W ∗ -independence has been settled only recently (Florig and Summers, 1997; Hamhalter, 1997, in press). First of all, W ∗ -independence always implies C ∗ -independence. Indeed, as the set of normal states is w ∗ -dense in the state space, W ∗ -independence implies the existence of common extensions for pairs of states forming w ∗ -dense set in the product of state spaces of local algebras. As the state space is compact in the w ∗ -topology we see that it is enough to imply C ∗ -independence. The set of normal states has different character then the state space (it is not compact, etc.). For this reason W ∗ -independence seems to be, at first glance, different from C ∗ -independence. Nevertheless, surprising result of Florig and Summers (1997) has shown that C ∗ -independence coincides with W ∗ independence for mutually commuting C ∗ algebras acting on separable Hilbert spaces. Representation on separable space is a basic ingredient of their proof. The question of validity of this theorem for general algebras remains open. In subsequent analysis of independence of subalgebras in general position, we have proved that W ∗ - and C ∗ -independence split up for all essentially noncommutative algebras.

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Theorem 2.1. (Hamhalter (in press)) Let M be a von Neumann algebra. The following conditions are equivalent: (i) M has an infinite-dimensional nonabelian direct summand. (ii) There are finite-dimensional von Neumann subalgebras A and B in M which are C ∗ -independent but not W ∗ -independent. Even, it can be proved that for any von Neumann algebra with infinitedimensional nonabelian direct summand, we can find two projections p, q ∈ M (of course not commuting) such that two-dimensional subalgebras A = sp{ p, 1 − p}, ∗

B = {q, 1 − q}

∗

are C -independent but not W -independent. Note that condition (i) in Theorem 2.1 is equivalent to the existence of state on M which is not Jauch–Piron. This fact is not a coincidence, the construction of a state which is not Jauch–Piron is an important step in constructing algebras A and B earlier. However, despite of this negative result, the position of C ∗ - and W ∗ -independence is more delicate. The following result shows that C ∗ -independence is very close to W ∗ -independence. Theorem 2.2. Let A and B be C ∗ -independent von Neumann subalgebras in a von Neumann algebra M. Then for any ε > 0 and for all normal states ϕ1 and ϕ2 on A and B, respectively, there is a normal state ϕ on M such that ϕ|A − ϕ1  < ε

and ϕ|B − ϕ2  < ε.

Therefore, C ∗ -independence is strong enough to ensure that the pairs of normal states on local algebras A and B that do admit common normal extension over M are norm dense in the product of normal state spaces. Because of ubiquitous error in measurement, Theorem 2.2 says that we cannot distinguish between C ∗ - and W ∗ -independence in a real situation. On the other hand, W ∗ - and C ∗ -independence considerably differ in the following respect. It was proved by H. Roos (1970) that for mutually commuting C ∗ -independent C ∗ algebras A and B in a C ∗ algebra C one can find for any pair of states ϕ1 on A and ϕ2 on B a state ϕ on C (called product state) such that ϕ(ab) = ϕ1 (a)ϕ2 (b)

for all a ∈ A, b ∈ B.

The product state embodies the well-known law for expectation value of product of independent variables. Although for mutually commuting von Neumann algebras on separable space C ∗ -independence coincide with W ∗ -independence, we can seldom find simultaneous product normal extension. For example, it is known that the von Neumann algebra M generated by mutually commuting von Neumann algebras A and B is isomorphic to the tensor product A⊗B whenever there is at least one normal product state on M with central support 1 (D’Antoni and Longo,

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1983; Takesaki, 1958). Let us now take a type III factor A on a separable Hilbert space H . Then the pair (A, A ), where A denotes the commutant of A in the algebra B(H ) of all bounded operators on H , is both C ∗ - and W ∗ -independent. The von Neumann algebra generated by A and A is a type I factor B(H ). As the tensor product of type III factors is again type III factor, we have that A⊗A is not isomorphic to the algebra generated by A and A . In this case, all pairs of normal states on A and A extend simultaneously to normal states on B(H ), but none of these extensions is a product state. On the other hand, any pair of normal states has common product extension. Let us remark that this situation is typical in quantum field theory. This advocates the use of nonnormal states in the C ∗ algebraic approach approach to quantum filed theory (Haag, 1996; Haag and Kastler, 1964). ACKNOWLEDGEMENT The author would like to express his gratitude to the Alexander von Humboldt Foundation for the support of his research the result of which are contained in this paper. He would also like to thank to the Grant Agency of the Czech Republic, Grant No. 201/00/0331, and Czech Technical University, Grant No. MSM 210000010, for supporting his research activity. REFERENCES Alfsen, E. M. and Schultz, F. W. (1976). Non-commutative spectral theory for affine functions spaces on convex sets. Memoirs of the American Mathematical Society 172. Amann, A. (1989). Jauch–Piron states in W ∗ -algebraic quantum mechanics. Journal of Mathematical Physics 28(10), 2384–2389. Baumg¨artel, H. and Wollenberg, M. (1992). Casual nets of operator algebras, Mathematical aspects of quantum field theory. Mathematische Monographien 80. Baumg¨artel, H. (1995). Operatoralgebraic Methods in Quantum Field Theory, Akademie Verlag, Berlin. Bunce, L. J. and Hamhalter, J. (1994). Jauch–Piron states on von Neumann algebras. Mathematische Zeitschrift 215, 491–502. Bunce, L. J. and Hamhalter, J. (1995). Countably additive homomorphisms between von Neumann algebras. Proceedings of the American Mathematical Society 123(11), 157–160. Bunce, L. J. and Hamhalter, J. (1996). Extension of Jauch–Piron states on Jordan algebras. Mathematical Proceedings of the Cambridge Philosophical Society 119, 279–286. Bunce, L. J. and Hamhalter, J. (2000). Jauch–Piron states and σ -additivity. Reviews in Mathematical Physics 12(6), 767–777. D’Antoni, C. and Longo, R. (1983). Interpolation by type I factors and the flip automorphism. Journal of Functional Analysis 51, 361–371. Florig, M. and Summers, S. J. (1997). On the statistical independence of algebras of observables. Journal of Mathematical Physics 38(3), 1318–1328. Goldstein, S., Luczak, A., and Wilde, I. (1999). Independence in operator algebras. Foundations of Physics 29(1), 79–89.

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